In this section we introduce the notion of twisting and very twisting scrolls, see Definition12.3. The main consequence of this definition occurs in Lemma12.5, which shows that a porcupine contained in a twisting scroll gives a moduli point t in(X/C/k) which is contained in aP1whose general point parameterizes a section. In other words, the P1 connects t, a point in a boundary stratum, to a moduli point in the interior, the
complement of the boundary divisor. It is important to have a geometric criterion for twisting and very twisting surfaces, which is the purpose of Lemma12.6. The main result of this section, Proposition12.12, is that a family X→C contains very twisting scrolls, if the geometric generic fiber Y has a very twisting scroll. The proof uses lemmas about gluing twisting and very twisting scrolls and then deforming to produce new twisting and very twisting scrolls. We also prove that there are good parameter spaces for twisting and very twisting scrolls. And we prove that there are good parameter spaces for those porcupines “penned” by a very twisting scroll.
Recall the definition of a ruled surface R in X→C with respect toLwas given at the start of Section9. The morphism R→X maps into the smooth locus of X→C, see Lemma9.1. Our first task is to show that (many) free sections lie on scrolls of free lines.
Lemma12.1. — In Situation8.1assume Hypothesis7.8holds for the restriction of f to some nonempty open set S⊂C. Let e0be an integer and let Z be an irreducible component ofe0(X/C/k) whose general point parametrizes a free section. Then there exists a nonempty open set U⊂Z such that every u∈U(k)corresponds to a section s such that (a) it is penned by a ruled surface R→X, and (b) any ruled surface R penning s has the property that all of its fibres are free lines in X.
Proof. — We are going to use Lemma 11.4. Note that although L is f -relatively ample, L may not be “absolutely” ample, i.e., it may not be ample on X. However, to prove the lemma we may replaceL byL⊗f∗N for some suitable very ample sheafN on C. Thus, without loss of generality, we may assume thatLis ample on the total space X. Similarly, ifN is sufficiently ample, then we may also assume that any rational curve in X of degree 1 is in a fibre of f :X→C.
Let T⊂T⊂X, Di be as in Lemma11.3. Because T⊂X has codimension≥2 there is a nonempty open U⊂Z such that every u∈U(k) corresponds to a section s which is disjoint from T. Furthermore, we then pick a nonempty open U ⊂U such every u∈U(k) corresponds to a section s that meets each irreducible component Di
transversally in smooth points, see [Kol96, II Proposition 3.7]. Finally, by part (1) of Hypothesis 7.8, we may find a further nonempty open U⊂U such that each section s :C→ X corresponding to a point of U meets the locus over which the evaluation morphism ev:M0,1(X/C,1)→X has irreducible rationally connected fibres.
Let s:C→X correspond to a k-point of U. By [GHS03] we can find a morphism g:C→M0,1(X/C,1)such that ev◦g=s. This corresponds to a ruled surface R→X which pens s. This proves (a). Next, let R be any ruled surface penning s. By Lemma9.1 it lies in the smooth locus of X→C. By Lemma 11.4and our choice of s all fibres of
R→C are free curves in X. This proves (b).
To define the notion of a twisting scroll, we introduce some notation. Let f :X→ C,Lbe as in Situation8.1. Consider a ruled surface h:R→X. The following
commu-tative diagram of coherent sheaves on R with exact rows
0 TR/C h∗TX/C NR/X 0
0 TR h∗TX NR/X 0
defines the coherent sheaf NR/X, which we call the normal bundle. Even when the mor-phism from R to X fails to be a closed immersion, the normal bundle is of fundamental importance cf. its use in [GHS03]. Moreover, if the sheafLis relatively very ample, which is satisfied in important cases, then the map h:R→X will be a closed immersion and NR/Xwill be a locally free sheaf.
Remark12.2. — There are two remarks about the case whereLis relatively ample, but perhaps not very ample. The first is that NR/X is always flat over C, even when it is not locally free over R. This follows as the maps TR/C|Rt →h∗TX/C|Rt are injective, and [Mat80, Section (20.E)]. In particular it has depth ≥1, its torsion is supported in codimension≥1, and any fibre meets the torsion locus in at most finitely many points.
This also shows that NR/X|Rt equals NRt/Xt, the normal bundle of the map Rt →Xt
(defined similarly). The second is that the deformation theory of the morphism R→X, with X held fixed, is given by H0(R,NR/X)(infinitesimal deformations) and H1(R,NR/X) (obstruction space). The deformation theory is described in [Ill71, Section 2.1.6, pp. 191–
192], especially [Ill71, Théorème 2.1.7]. In particular, the cotangent complex of h:R→ X is given by h∗1X→1R which is quasi-isomorphic to h∗1X/C→1R/C. It follows that Exti(h∗1X/C→1R/C,OR)=Hi−1(R,NR/X)as usual.
Definition 12.3. — In Situation8.1. Let R→X be a ruled surface in X and let D be a Cartier divisor on R. For every nonnegative integer m, we say(R,D)is m-twisting if
(1) the complete linear system|D|is basepoint free, (2) the cohomology group H1(R,OR(D))is 0, (3) D has relative degree 1 over C,
(4) the normal bundle NR/Xis globally generated, (5) H1(R,NR/X)equals 0, and
(6) we have H1(R,NR/X(−D−A))=0 for every divisor A which is the pullback of any divisor on C of degree≤m.
This only depends on the Cartier divisor class of the Cartier divisor D.
This is a “relative” definition—it is defined with respect to the morphism f . An important special case is an “absolute” variant, see Definition12.7below.
Suppose we are in Situation8.1and suppose that(R,D)is an m-twisting scroll in X/C. By assumption |D| is nonempty, base point free, and of relative degree 1. Hence a general element is smooth and defines a section σ :C→R. We will often say “let (R, σ )be an m-twisting scroll” to denote this situation. Having chosenσ we can think of R→X as a family of stable 1-pointed lines. Let g=g(R,σ ):C→M0,1(X/C,1)denote the associated morphism.
Lemma12.4. — In the situation above:
(1) The image of g=g(R,σ )lies in the unobstructed (and hence smooth) locus of the morphism ev:M0,1(X/C,1)→X.
(2) We have H1(C,g∗Tev(−A))=0 for every divisor A of degree≤m on C.
(3) Let:M0,1(X/C,1)→M0,0(X/C,1)be the forgetful morphism (which is smooth).
Then g∗Tis globally generated, and H1(C,g∗T)=0.
(4) The section h◦σ :C→X is free, see Definition4.7.
Proof. —
(1) The fact that NR/X is globally generated implies that for each t ∈C(k) the fibre Rt → X is free. This follows upon considering the exact sequences 0→TR|Rt →h∗TX|Rt →NR/X|Rt →0 (exact by flatness of NR/Xover C), and the fact that TR|Rt is globally generated. This implies the image of g is in the unobstructed locus for ev.
(2) Letπ :R→C be the structural morphism. The pullback of the relative tan-gent bundle Tev by g is canonically identified with π∗NR/X(−σ ). This is true because the normal bundle of a fibre Rt →X is an extension of the restric-tion NR/X|Rt by a rank 1 trivial sheaf on Rt ∼=P1. The assumptions of Defi-nition 12.3 imply that R1π∗NR/X(−σ )=0. Hence by the Leray spectral se-quence for NR/X(−σ −π∗A), H1(C,g∗Tev(−A))=H1(R,NR/X(−σ−π∗A)).
Thus we get the desired vanishing from the definition of twisting scrolls.
(3) The pullback g∗T is canonically identified withσ∗OR(σ ). The global gener-ation of the sheaf g∗Tis therefore a consequence of the base point freeness of OR(σ )of Definition12.3. The vanishing of H1(C,g∗T)follows on consider-ing the long exact cohomology sequence associated to 0→OR→OR(σ )→ σ∗σ∗OR(σ )→0 and the vanishing of H1(R,OR(σ ))in Definition12.3.
(4) Consider the exact sequence σ∗TR/C → σ∗h∗TX/C → σ∗NR/X → 0. Note that σ∗TR/C =σ∗OR(σ ). By Definition 12.3 both σ∗OR(σ ) and σ∗NR/X
are globally generated. Thus by the exact sequence σ∗h∗TX/C is glob-ally generated. This is obvious if the first map is zero, otherwise the ex-act sequence is a short exex-act sequence, and in the previous paragraph we showed that H1(C, σ∗OR(σ ))=0. There is an exact sequence NR/X(−σ )→ NR/X→σ∗σ∗NR/X→0. By Definition 12.3, H1(R,NR/X)=0. The group
H2(R,NR/X(−σ )) vanishes as its Serre dual HomR(NR/X(−σ ), ωR) is zero (hint: consider restriction to fibres). Together these imply that H1(C, σ∗NR/X)=0. Thus the first exact sequence of this paragraph implies that H1(C, σ∗h∗TX/C)=0. Thus(h◦σ )∗TX/Cis globally generated with trivial H1and we conclude that h◦σ is 1-free, i.e., free.
The following innocuous looking lemma is why we introduce twisting scrolls. It will eventually show that for every canonical irreducible component of the moduli space of stable sections with sufficiently positive degree, for the associated “boundary” stratifica-tion of the moduli space (according to the dual graph of the domain curve), a general point of a particular boundary stratum is connected by aP1to a point in the “interior”, i.e., the open complement of the boundary divisor.
Lemma12.5. — In Situation8.1, let(R, σ )be m-twisting with m≥1. Let t∈C(k). The stable mapσ (C)∪Rt→X defines a nonstacky unobstructed point of(X/C/k)which is connected by a rational curve in(X/C/k)to a free section of X→C.
Proof. — The fact that the point is nonstacky comes from the fact that sections and lines have no automorphisms. The fact that the point is unobstructed follows from Lemma 12.4above. Consider the linear system |σ +Rt|. Since H1(C,OR(σ ))=0 the map H0(R,OR(σ +Rt))→H0(Rt,ORt(1)) is surjective. This implies that |σ +Rt| is base point free. A general member of|σ+Rt|is a sectionσ:C→R. It is trivial to show that(R, σ)is(m−1)-twisting. Hence by Lemma12.4we see thatσis free. The result is clear now by considering the pencil of curves on R connectingσ +Rt toσ. It is useful to have a criterion that guarantees the existence of a twisting sur-face. In particular, we would like a condition formulated in terms of the map g:C→ M0,1(X/C,1). We do not know a good way to do this unless g(C)=0.
Lemma 12.6. — In Situation 8.1assume C=P1. Let g:P1→M0,1(X/P1,1) be a section. Let m≥1. Assume we have:
(1) The pullback g∗T has degree≥0.
(2) The image of g is contained in the unobstructed locus of ev.
(3) The cohomology group H1(P1,g∗Tev(−m))is zero.
(4) The composition ev◦g:P1→X is a free section of X→P1. Then the associated pair(R, σ )is a m-twisting scroll in X.
Proof. — We will use the identifications of g∗T = σ∗OR(σ ) and g∗Tev = π∗NR/X(−σ ) from the proof of Lemma 12.4. Note that R is a Hirzebruch surface, and H1(R,OR)=H2(R,OR)=0. Combined with the fact that σ2=deg(g∗T)≥0
we see that |σ| is base point free. Also, during the course of the proof we may assume that σ is general in its linear system on R. In particular this means that the section 1∈ (R,OR(σ ))is regular for the coherent sheaf NR/X. (Note that this is automatic in the case, which always holds in practice, that R→X is a closed immersion.)
The fact that ev◦g :P1 →X is 1-free means that g∗ev∗TX/P1 is globally gen-erated. The fact that H1(P1,g∗Tev)=0 means that any infinitesimal deformation of the morphism ev ◦g :P1 → X can be followed by an infinitesimal deformation of g :P1 → M0,1(X/C,1). In terms of the pair (R, σ ) this means that the image of α:H0(R,NR/X)→H0(P1, σ∗NR/X)contains the image ofβ:H0(P1, (ev◦g)∗TX/P1)→ H0(P1, σ∗NR/X). In this way we conclude that NR/X is at least globally generated over the image ofσ.
This weak global generation result in particular implies that R1π∗NR/X(−σ )=0, and R1π∗NR/X=0; we can for example see this by computing the cohomology on the fibres. Thus we see that H1(R,NR/X(−σ −π∗A))=H1(P1,g∗Tev(−A)). This gives us the vanishing of the cohomology group H1(R,NR/X(−σ −π∗A)) for any divisor A of degree≤m on P1. We also get H1(R,NR/X)=H1(P1, π∗NR/X), and an exact sequence 0→π∗NR/X(−σ )→π∗NR/X→σ∗NR/X→0. The first sheaf being identified with g∗Tev
and the second being globally generated we conclude that H1(R,NR/X)=0. Note that, with m>0 this argument actually also implies that H1(R,NR/X(−Rt))=0 for any t∈ C(k). At this point, what is left, is to show that NR/X is globally generated. It is easy to show that a coherent sheaf on P1 which is globally generated at a point is globally generated. Since m≥ 1 the map H0(R,NR/X)→H0(Rt,NR/X|Rt) is surjective by the vanishing of cohomology we just established. Combined these imply that NR/Xis globally
generated.
Here is the definition we promised above.
Definition12.7. — Let k be an algebraically closed field of characteristic 0. Let Y be projective smooth over k, and letLbe an ample invertible sheaf on Y. A scroll on Y is a scroll for the morphism pr1:P1×Y→P1, i.e., it is given by morphismsP1←R→Y such that (a) R is a smooth projective surface, (b) all fibres Rtof R→P1are nonsingular rational curves, and (c) the induced maps Rt→Y are lines in Y. Let R be a scroll in Y and suppose D⊂R is a Cartier divisor. For every nonnegative integer m, we say(R,D)is an m-twisting scroll in Y if the diagram
R P1×Y
pr1
P1 P1
and the divisor D form an m-twisting scroll with respect to the invertible sheaf pr∗2L. If m is at least 2, an m-twisting scroll will also be called a very twisting scroll.
The main observation of this section is that if X→C is as in Situation8.1, and Hypothesis7.8holds over an open part of C, and if the geometric generic fibre has a very twisting scroll, then X→C also has very twisting scrolls.
Lemma 12.8. — In Situation 10.1, assume that Y has a very twisting scroll. Then there exist m-twisting scrolls (π :R→P1,h :R→Y, σ ) such that π∗OR(D) and π∗NR/P1×Y are ample locally free sheaves on P1 with m arbitrarily large. Moreover, for a sequence(Ze)e≥e0 as defined in Section10, we can further arrange it so that h◦σ corresponds to an unobstructed point of one of the irreducible components Ze.
Proof. — First we have a simple construction which begins with m-twisting scrolls for an integer m, and produces m-twisting scrolls for a larger integer m. Let(R,D)be an m-twisting scroll on Y. Choose a sectionσ of R in|D|and think of the associated morphism g:P1→M0,1(Y,1)as in Lemmas12.4and12.6. In fact Lemma 12.4implies that the morphism g satisfies the assumptions of Lemma12.6. It is clear that the assumptions of Lemma 12.6 hold on an open subspace of Mor(P1,M0,1(Y,1)). Also, a morphism g satisfying those assumptions is free. Given two morphisms gi:P1→M0,1(Y,1), i=1,2 corresponding to mi-twisting surfaces with g1(∞)=g2(0) there exists a smoothing (see [Kol96, II Definition 7.1, Theorem 7.6]) of g1∪g2:P1∪∞∼0P1→M0,1(Y,1)whose general fibre is a morphism g:P1→M0,1(Y,1). As explained above, such a morphism gives aP1-bundle R overP1, a Cartier divisor D on R (the “marked point” divisor), and a morphism from R toP1×Y compatible with the projection toP1. The hypotheses in Definition12.3are each open in families. Thus using that g1is m1-twisting and that g2is m2-twisting, it follows directly that for the general smoothing g, also g is an(m1+m2−1) -twisting scroll.
By hypothesis, there exist 2-twisting scrolls. Using the construction above, we con-clude that there exist 3-twisting scrolls. Also if (R→P1,R→Y,D)is m-twisting with m≥3, then (R→P1,R→Y,D+R0)is (m−1)-twisting and has the property that π∗OR(D+R0)is ample.
Fix m≥3 such that an m-twisting scroll exists. Consider a large integer N and con-sider maps gi:P1→M0,1(Y,1), i=1, . . . ,N each corresponding to an m-twisting scroll (πi:Ri→P1,hi:Ri→X, σi), and such that gi(∞)=gi+1(0)for i=1, . . . ,N−1. We may assume that all the free rational curves hi◦σilie in the same irreducible component Z ofM0,0(Y,e0), for example by taking the same twisting scroll for each i (with coordi-nate onP1reversed for odd indices i). We may also assume that N≥E−e0where E is as in assertion (6) in Section10. Since each gi is free we can find a smoothing of the stable map g1∪. . .∪gN:P1 tion the ith component of the initial chain.
Let (π :R→P1,h:R→Y, σ ) correspond to a general point of the smoothing, and let t1, . . . ,tN∈P1(k)be the values of the sections zi. We claim that(R→P1,h:R→
Y, σ +
Rti) is a twisting surface which has two of the three desired properties, i.e., π∗OR(σ +
Rti)is ample and the section curve can be chosen to be an unobstructed point of some Ze. The third property, ampleness ofπ∗NR/X, will require more work.
First, since(R, σ )is Nm-twisting we see immediately that(R, σ+
Rti)is N(m− 1)twisting.
Second, the Cartier divisorσ +
Rti is a deformation of a divisor on the surface R1 because(R, σ )is a twisting scroll. By construction our comb is a “partial smoothing” of the tree of rational curves
Again since all fibres of twisting scrolls are free lines, and since eachσi is free this stable map is unobstructed. At this point we may apply Lemma10.3to conclude that this stable map is in Ze. is very free, see Section10 (6). At this point consider the exact sequence of locally free sheaves onP1
0→π∗NR/P1×Y(−σ)→π∗NR/P1×Y→σ∗NR/P1×Y→0.
The sheaf on the left hand side is ample because(R, σ)is N(m−1)-twisting. Combined with the surjective map(σ)∗TY→σ∗NR/P1×Yand the ampleness of(σ)∗TYthis proves
the result.
In order to formulate the next lemma, we need a definition. A very twisting scroll (π :R→P1,h:R→Y, σ )as in the lemma above is called wonderful if the pushforward sheaf π∗NR/P1×Y (which is locally free by Remark 12.2) is ample, if π∗OR(σ ) is ample,
and if the section h◦σ belongs to the canonical irreducible component Ze defined in Section10for some integer e (of course the integer e equals the degree of h◦σ).
Lemma 12.9. — In Situation 8.1, assume that Hypothesis7.8holds over a nonempty open S⊂C. Assume that for some t∈C(k)the fibre Xt has a very twisting scroll. Then there exists a family of wonderful, very twisting scrolls, in the following precise sense. There exist
(1) a smooth variety B over k, (2) a flat morphism t:B→C,
(3) a smooth projective family of surfacesR→B, (4) a morphismπ :R→P1,
(5) a morphism h:R→X such that f ◦h=t, and
(6) a morphismσ :P1×B→Rover B such thatπ ◦σ =pr1. These data satisfy:
(1) for each b∈B(k) the fibre (πb:Rb→P1,hb:Rb→Xt(b), σb) is a wonderful very twisting scroll in Xt(b), and
(2) the image of the map
fib0:B−→M0,1(X/C,1),
which assigns to b∈B(k)the 1-pointed free line σb(0)∈πb−1(0)→Xt(b) contains a nonempty open V⊂M0,1(X/C,1).
Remark12.10. — The formulation above is just one possible formulation of “family of wonderful very twisting scrolls”. The important part of the lemma is once there is a single such a scroll, then there are many.
Proof. — Let (P1 ← R → Xt,D) be a scroll in a fiber. As in the proof of Lemma 12.8, we consider this as a morphism P1 →M0,1(P1×Xt,1)⊂M0,1(P1× X,1). And this gives a point in Mor(P1,M0,1(P1×X,1)). For a suitably chosen R, B will be an open subset of Mor(P1,M0,1(P1×X,1))containing this point.
Let (P1←R→Xt,D)be an m-twisting scroll in a fibre. There are no obstruc-tions to deforming the morphism R→P1×Xt because H1(R,NR/P1×Xt)=0, see Re-mark12.2. Also there are also no obstructions to deforming the morphism R→P1×X:
the normal bundle of this morphism is a direct sum of NR/X and a trivial summand TtC⊗κ(t)OR∼=OR, where the splitting is induced by the derivative df of the morphism f :X→C (which is constant on R by hypothesis). Since H1(R,NR/P1×Xt)equals 0 and since also H1(R,OR)equals 0 (as R is a rational surface), also H1(R,NR/P1×X)equals 0.
A deformation of R→P1×X is a scroll R→P1×X which is also contained in a fiber P1×Xt of f over some point t of C. Also sinceOR(D)is globally generated on R and since h1(R,OR(D))equals 0, we can deform D to a divisor D on R. Since the hypothe-ses of Definition 12.3are open conditions, a general deformation (R,D) of (R,D)is
still m-twisting. And since the normal bundle of R→P1×X is globally generated, we see that we may deform R→P1×Xt to a morphism R →P1×Xt with t general and R→Xt passing through a general point of Xt. So this gives very twisting scrolls in fibres over S. Since Hypothesis 7.8holds over S, these scrolls in fibres over S are in Situation10.1. Thus we may apply Lemma12.8to these scrolls. Therefore we may now assume that R is a wonderful twisting scroll.
Moreover, let σ :P1 →R be a section of R→P1 such that D∼σ are ratio-nally equivalent. Pick a point p∈P1(k). Since m>0 the map H0(R,NR/P1×Xt(−σ ))→ H0(Rp,NRp/Xt(−σ (p)))is surjective. We see that given any infinitesimal deformation of the pointed map (Rp, σ (p))→(Xt,h(σ (p))) (from a pointed line to Xt pointed by the image of the point) in Xt we can find an (unobstructed) infinitesimal deformation of R→P1×Xt that induces it. Combined with the fact, proven above, that we can pass a deformation of R through a general point of X this shows that we can deform R→Xt
such that a given fibre of R→P1is a general line in X/C.
Now we consider the point of Mor(P1,M0,1(P1×X,1)) corresponding to this scroll, and we define B to be a suitable open neighborhood. Finally, what is left is to show that in a family of morphisms of surfaces R→P1×X the locus where the surface is a wonderful m-twisting scroll is open. This follows from semi-continuity of cohomology and can safely be left to the reader, although a very similar and more difficult case is handled
in Lemma12.11below.
In Situation8.1, let R→X be a ruled surface all of whose fibres are free lines in X/C. Let D be a Cartier divisor on R of degree 1 on the fibres of R→C. Let t1, . . . ,tδ∈
Note that there is a Cartier divisor Don Rwhose restriction to R is rationally equivalent to D and whose restriction to each Siis rationally equivalent to Di. Moreover this Cartier divisor is unique up to rational equivalence.
In the following we are interested in smoothings of situations as above. This means that we have an irreducible variety T over k, and a commutative diagram of varieties
R=R∪
satisfying the following conditions: (1)C→T andR→T are flat and proper, (2) every fibreCt ofC →T over t∈T(k)is a nodal curve of genus g(C)andCt→C has degree 1
(see discussion in Section6), (3) the morphism R→C is smooth, (4) every fibreRt→ Ct ×CX is a ruled surface over every irreducible component of Ct, (5) for some point 0∈T(k)the fibreR0→C0×CX is isomorphic to our map h:R→C×CX above, and finally (6) for some point t∈T(k)the fibreCt=C. In addition we assume given a Cartier divisorDonRrestricting to the divisor Don R=R0.
Lemma12.11. — In the situation above.
(1) If(R,D)is m-twisting and(Si,Di)i=1...δare mi-twisting, with mi≥2, then for t∈T(k) general the ruled surface(Rt,Dt)in X is(m+δ)-twisting.
(2) Let ni, i=1,2,3 be integers such that for all Cartier divisors A of degree on C we have (a) deg(A)≥n1⇒H1(C, (π∗NR/X)(A))=0,
(b) deg(A)≥n2⇒H1(C, (π∗NR/X(−D))(A))=0, and (c) deg(A)≥n3⇒H1(C, (π∗OR(D))(A))=0.
If (Si,Di)i=1...δ are wonderful very twisting scrolls andδ≥max{n1+1,n2+1,n3}, then for t∈T(k)general the ruled surface(Rt,Dt)in X is(δ−n3)-twisting.
Proof. — The main part is (2). Part (1) follows by applying the proof of (2) and the proof of Lemma 12.8. Regarding the proof of (1), we would like to remark that the increase in the twisting comes from the fact that the sheaves(Si→P1)∗ NSi/P1×Xti(−Di) are ample vector bundles on P1: they are locally free by Remark 12.2, and ampleness follows from the hypothesis that each mi≥2, combined with [Kol96, II Lemma 7.10.1].
We begin the proof of (2). Because the morphismR→C is smooth we can define NR/Xby the short exact sequence
0→TR/C→(R→X)∗TX/C→NR/X→0.
As before this is a sheaf on R which is flat over C. For every point c∈C(k) lying over t∈C(k)the restriction of NR/Xto the fibreRc is the normal bundle of the lineRc→Xt
(compare with Remark 12.2). For all points c∈C(k) lying over 0 the sheaf NR/X|Rc is globally generated. Hence after replacing T by an open subset containing 0 we may assume this is true for all points c.
Consider the coherent sheaves ofOC-modulesE1=π∗OR(D),E2=π∗NR/X and E3 =π∗NR/X(−D). By the above the corresponding higher direct images are zero (as H1 of a globally generated sheaf onP1 is zero). The semicontinuity theorem impliesEi
is a locally free sheaf on C. Let U⊂T be a nonempty open such that Ct=C for all
is a locally free sheaf on C. Let U⊂T be a nonempty open such that Ct=C for all